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APPLICATIONS  OF  ALGEBRA 


DEALING   WITH 


B      AUTOMOBILES 

FOR   USE    IN   CONNECTION  WITH   THE   FIRST   YEAR's 
WORK    IN    ALGEBRA 


BY 

THIRMUTfflS  BROOKIVIAN  AND  OTHERS 

MEMBERS   OF   THE   TEACHERS    CLASS    IN    APPLIED   MATHEMATICS 
UMVEBSITY    PF    CALIFORNIA,    Sl'MMEK    SE3SIOX,    1913 


REVISED    19U.  1915 


=1 


CHARLES  SCRIBNER'S  SONS 

NEW   YORK  CHICAGO  BOSTON 


APPLICATIONS  OF  ALGEBRA 


DEALING   WITH 


AUTOMOBILES 


FOR   USE   IN   CONNECTION  WITH   THE   FIRST   YEAr's 
WORK    IN    ALGEBRA 


BY 

THIRMUTfflS\BR0OKMAN  AND  OTHERS 

MEMBERS   OF   THE   TEACHERS    CLASS    IN'    APPLIED   MATHEMATICS 
UNIVERSITY   OF  CALIFORNIA,   SUMMER   SESSION,    1913 


REVISED    1914,  1915 


i  CHARLES  SCRIBNER'S  SONS 

y     11  NEW  YORK  CHICAGO  BOSTON 


CAJORI 


Copyright,  1916,  bt 
CHARLES  SCRIBNER'S  SONS 


CONTRIBUTORS 


Thirmuthis  Brookman 
O.  W.  Baird  .... 
Geo.  T.  Brooks  .  .  . 
Lilly  E.  Burkhardt  . 
Alex.  R.  Craven  .  . 
Flora  E.  Crowley  .  . 
Laura  Gilbert  .  .  . 
Catherine  Lamberson 
F.  J.  Lawrence  .  .  . 
B.  a.  Lindsay  .  .  . 
Kate  Mitchell  Meek 
George  E.  Mercer 
Emily  G.  Palmer  .  . 
Georgia  M.  Simon  .  . 
Charles  E.  Taylor  . 
Anna  G.  Wright     .     . 


San  Francisco,  Calif. 

High  School,  Nome,  Alaska. 

High  School,  Hutchinson,  Kansas. 

San  Francisco,  Calif. 

Lowell  High  School,  San  Francisco,  Calif. 

High  School,  Williams,  Calif. 

High  School,  Corona,  Calif. 

Washington  High  School,  Portland,  Ore. 

High  School,  Inglewood,  Calif. 

High  School,  Sparks,  Nev. 

High  School,  South  Pasadena,  Calif. 

High  School,  Palo  Alto,  Calif. 

High  School,  Salem,  Ore. 

High  School,  Tuolumne,  Calif. 

High  School,  Berkeley,  Calif. 

Mayfield,  Calif. 


Digitized  by  the  Internet  Arciiive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/applicationsofalOObroorich 


PREFACE 

The  purpose  of  this  booklet  is  to  vitalize  first-year  al- 
gebra by  applying  it  to  objects  with  which  the  pupils  may 
readily  become  famihar  in  daily  life. 

Among  the  many  machines  whose  operations  can  be 
fairly  comprehended  by  those  whose  knowledge  is  limited  to 
elementary  algebra,  none  seem  of  more  far-reaching  interest 
and  importance  than  the  automobile.  These  pages,  therefore, 
develop  the  simpler  algebraic  formulas  used  in  the  operation 
of  automobile  engines,  in  the  transmission  of  speed,  and  in 
problems  dealing  with  automobiles  on  the  road. 

The  algebra  used  in  the  solution  of  these  appHcations  is  the 
linear  equation,  direct  proportion,  and  a  Umited  knowledge 
of  pure  quadratics.  The  applications  of  elementary  algebraic 
principles  herein  are  similar  to  those  developed  throughout 
the  first  year's  work  in  algebra  in  Brookman's  Practical 
Algebra  for  Beginners.  It  is  expected  that  such  problems 
of  real  life,  which  are  of  a  nature  to  enlist  the  ready  interest 
of  many  pupils,  will  replace  the  difficult  manipulations  of 
abstract  symbols  included  in  several  of  the  current  texts, 
and  will  also  give  more  real  significance  to  the  algebraic  equa- 
tion. It  is  hoped  that  this  booklet  will  encourage  the  beginner 
to  enter  into  the  study  of  algebra  with  alertness  and  keen 
interest,  because  he  reahzes  that  it  gives  him  mastery  over 
the  practical  formulas  needed  in  actual  experience. 

The  writers  wish  to  acknowledge  their  indebtedness  to  all 
who  have  given  them  suggestions  and  practical  information, 
and  especially  to  Mr.  Alden  McElrath  of  the  Oakland  head- 
quarters of  the  Cadillac  Motor  Company. 

San  Franciscx),  June,  1916. 


CONTENTS 

CHAPTER  PAGE 

I.    AUTOMOBILE  ENGINES 1 

1.  General  Description 1 

2.  The  Gasolene-Engine 4 

3.  Problems  on  a  One-Cylinder  Engine       .      .  6 

4.  Horse-Power  of  Gas-Engines 7 

5.  Problems  on  Piston  Pressure   .....  8 

6.  Formula  for  Energy  per  Minute  ....  10 

7.  Problems  on  Revolutions  of  the  Fly- Wheel  .  11 

8.  Problems  on  Horse-Power 12 

n.    TRANSMISSION  OF  SPEED 14 

9.  Gears  in  Mesh 14 

10.  Problems  on  Gears 15 

11.  High-Speed  Transmission 16 

12.  Problems  on  High-Speed  Transmission     .      .  18 

13.  Gear-Box  and  Gears  in  Neutral  ....  19 

14.  Intermediate  Speed 22 

15.  Problems  on  Intermediate  Speed  ....  26 

16.  Problems  on  Low  Speed 29 

17.  Reverse  Speed 31 

m.    AUTOMOBILES  IN  MOTION 34 

18.  Road  Problems 34 

19.  Formulas  Concerning  Automobiles     ...  39 


APPLICATIONS  OF  ALGEBRA 

DEALING  WITH  AUTOMOBILES 


FOR  USE  IN   CONNECTION  WITH   THE   FIRST  YEAR  S 
WORK   IN  ALGEBRA 


CHAPTER  I 
AUTOMOBILE  ENGINES 

1.  General  Description.  The  relation  of  the  different 
parts  of  an  automobile  may  be  seen  in  the  following  pic- 
ture of  one  of  the  latest  types  of  machines.  This  descrip- 
tion, with  slight  variations,  applies  to  all  popular  makes  of 
cars.  The  parts  italicized  in  the  description  should  be  lo- 
cated in  the  following  diagram. 

The  power  which  runs  the  car  is  generally  obtained  from 
gasolene  or  electricity.  This  book  will  consider  only  those 
machines  which  are  run  by  gasolene.  The  motive  power  of 
such  machines  is  generated  in  cylinders  located  in  front  of 
the  car.  The  movement  of  the  piston  in  each  cylinder 
helps  to  turn  the  crank-shaft,  which  is  beneath  the  cylinders. 
(The  crank-shaft  does  not  appear  in  the  drawing.) 

The  crank-shaft  and  the  fly-wheel  (usually  behind  the 
engine)  are  securely  connected  and  so  revolve  together. 
Into  the  hollowed  portion  of  the  fly-wheel  is  fitted  a  cone 
clutch,  by  means  of  which  the  motion  of  the  fly-wheel  may 
be  imparted  to  the  machine.      When  the  cone  clutch  is 

1 


Cylinders 

Flywheel 

Cone  Clutch 

Main    Tran6^ 
mission  Shaft 

Gear  Box 


— Rear  Wheel 

^Rear  Axle 
Bevel  Gear 


DIAGRAM    SHOWING    OPERATING    MECHANISM 


Cylinder)! 
\  of  gas- 
I  engine 


Fly-wheel 


•e  Cfme  diUch 


Main  transmiasion- 
shajl 


CADILLAC   OPERATING   MECHANISM 
3 


4  APPLICATIONS   OF   ALGEBRA 

disconnected  or  the  hand-lever  is  in  neutral  position  the  en- 
gine may  run  while  the  automobile  is  standing  still.  This 
is  called  "idling."  From  the  cone  clutch  extends  the  main 
transmission-shaft  in  two  parts  through  the  gear-box  to  the 
bevel-gears,  which  turn  the  rear  axle  of  the  machine.  Some 
machines  have  the  multiple  disk  or  some  other  type  of  clutch 
in  place  of  the  cone.  (If  possible  pupils  should  see  the  oper- 
ating mechanism  of  several  different  automobiles.) 

2.  The  Gasolene-Engine.  The  engine  is  the  most  vital 
and  also  the  most  intricate  part  of  the  automobile,  and  an 
understanding  of  its  construction  and  workings  is  of  the 
utmost  importance.  Most  motor-cars  are  equipped  with  en- 
gines of  four,  six,  or  eight  cylinders,  while  the  simplest  form 
of  engine  has  but  one. 


IV 


E.V. 


LOADING  COMPRESSING  EXPLODING  EXHAUSTING 

CYLINDER    SHOWING    FOUR    POSITIONS    OF    PISTON 

The  cylinder  is  closed  at  the  top  and  fitted  with  a  piston 
(P)  connected  with  a  rod  {R)  to  the  crank-shaft  {C.S.). 
The  accompanying  picture  of  one  type,  the  four-cycle  gas- 
engine,  shows  four  positions  of  the  piston.  Above  the 
piston  is  the  combustion  chamber   (CC),  where  there   are 


AUTOMOBILE  ENGINES  5 

two  valves,  an  inlet  valve  (/.F.)  and  an  exhaust  valve  {E.V.). 
Find  these  parts  in  the  accompanying  illustration. 

/,  II,  III  and  IV  show  four  different  positions  of  the 
piston  in  a  single  cylinder. 

In  /,  if  the  machine  is  in  motion,  the  piston  is  approaching 
the  bottom  of  the  cylinder.  As  the  piston  moves  down- 
ward the  inlet  valve  (I.V.)  opens  to  admit  a  mixture  of 
gasolene- vapor  and  air,  the  explosion  of  which  is  to  furnish 
the  motive  power.  This  valve  closes  automatically  when  the 
piston  is  near  the  bottom  of  the  stroke. 

In  II  the  piston  is  approaching  the  top  of  the  cylinder  and 
compressing  the  gas  in  the  combustion  chamber  to  a  frac- 
tion of  its  former  volume. 

In  III  the  dotted  line  shows  the  highest  position  which 
the  piston  reaches.  Near  this  point  an  electric  spark  from 
the  spark-plug  (S.P.)  explodes  the  compressed  gas  in  the 
cylinder  and  the  piston  is  immediately  forced  downward 
by  the  expansion  of  the  gas. 

In  IV  the  piston  has  again  reached  its  lowest  position,  and 
as  it  moves  upward  the  exhaust  valve  {E.V.)  opens  and  the 
piston  forces  out  the  spent  gas.  As  the  piston  reaches  the 
top  the  exhaust  valve  closes,  the  inlet  valve  opens,  the  piston 
returns  again  to  the  position  shown  in  /,  and  the  cycle  is 
completed. 

In  order  to  obtain  increased  power  for  running  the  ma- 
chine, the  one-cylinder  engines  have  been  replaced  by  those 
containing  more  cylinders.  As  the  number  of  cylinders 
increases,  the  number  of  explosions  in  the  engine  to  every 
revolution  of  the  fly-wheel  also  increases  and  produces  a 
more  continuous  power  and  a  more  smoothly  running  engine. 
However,  the  ratio  between  the  number  of  explosions  in  one 
cylinder  and  the  revolutions  of  the  crank-shaft  remains  the 


6  APPLICATIONS  OF  ALGEBRA 

same.  In  the  four-cylinder  car  four  explosions,  one  in  each 
cylinder,  cause  two  revolutions  of  the  fly-wheel;  likewise,  in 
the  six-cylinder  car,  six  explosions,  one  in  each  cylinder, 
cause  only  two  revolutions  of  the  fly-wheel. 

Questions 

1.  As  the  piston  passes  from  one  extreme  position  to 
another  show  the  number  of  degrees  through  which  the 
point  A  on  the  crank-shaft  passes. 

2.  As  the  piston  completes  the  four  phases  of  one  cycle, 
show  that  point  A  on  the  crank-shaft  passes  through  720°. 

3.  During  the  four  phases  of  one  cycle  how  many  revo- 
lutions does  the  fly-wheel  make.^^  How  many  explosions 
does  the  one-cylinder  engine  make? 

4.  How  many  revolutions  would  the  fly-wheel  make  for 
10  explosions  in  a  one-cylinder  engine  ?  How  many  for  120 
explosions  ? 

5.  When  the  fly-wheel  turns  40  times,  how  many  explo- 
sions occur  in  the  cylinder  ? 

3.  Problems  on  a  One-Cylinder  Engine.  In  the  follow- 
ing problems  consider  the  engine  as  having  one  cylinder 
only. 

1.  (a)  If  a  gas-engine  makes  420  explosions  per  minute, 
how  many  revolutions  will  the  fly-wheel  make  in  the  same 
time? 

(b)  How  many  revolutions  will  the  fly-wheel  make  in 
5  minutes  ? 

2.  If  the  fly-wheel  of  a  gas-engine  makes  1200  revolu- 
tions per  minute,  how  many  explosions  will  there  be? 

3.  (a)  If  a  gas-engine  of  a  motor-cycle  makes  484  ex- 
plosions per  minute,  and  the  fly-wheel  turns  11  times  while 


AUTOMOBILE  ENGINES  7 

the  rear  wheel  turns  3  times,  how  often  will  the  rear  wheel 
turn  in  a  minute? 

(b)  If  the  circumference  of  the  rear  wheel  of  a  motor- 
cycle is  8  ft.,  how  far  does  it  travel  in  a  minute? 

(c)  How  long  will  it  take  to  travel  half  a  mile  ? 

4.  (a)  The  circumference  of  the  rear  wheel  of  a  motor- 
cycle is  7^  ft.  If  the  rear  wheel  makes  3  revolutions  to  11 
of  the  fly-wheel,  how  many  revolutions  will  the  fly-wheel 
make  in  travelling  one-half  mile  ? 

(b)  How  many  explosions  will  the  engine  make  in  trav- 
elling 3  miles? 

5.  (a)  The  diameter  of  the  rear  wheel  of  a  motor-cycle 
is  2.5  ft.  The  fly-wheel  turns  55  times  while  the  rear  wheel 
is  turning  16  times.  How  many  feet  will  the  motor-cycle 
have  moved  when  the  engine  has  made  2000  explosions  ? 

(b)  If  the  fly-wheel  makes  1500  revolutions  per  minute, 
what  will  be  the  speed  in  miles  per  hour  ? 

4.  Horse-Power  of  Gas-Engines.  The  explosion  in  the 
cylinder  produces  a  mean  effective  pressure  of  from  40  to  70 
pounds  per  square  inch  in  all  directions. 

To  obtain  the  number  of  foot-pounds  of  energy  developed 
by  a  single  explosion  in  a  cylinder,  three  things  must  be 
considered : 

P,  the  average  pressure  of  the  gas  in  pounds  per  square 
inch. 

a,  the  area  of  the  piston  in  square  inches. 

I,  the  length  of  the  stroke  in  inches. 

If  d  is  the  diameter  of  the  piston  or  cylinder,  the  area  of 

the  piston  is  obtained  by  the  formula 

Tvd?-  3.1416(^2        ^„r,,, 

a  =  — r-  or  a  =  -, =  .7854^^ 

4  4 

Since  the  area  of  a  circle  equals  the  product  of  t  and  the 

square  of  the  radius,  explain  how  this  formula  is  obtained. 


8  APPLICATIONS  OF  ALGEBRA 

5.  Problems  on  Piston  Pressure. 

1.  If  the  diameter  of  a  piston  is  4  in.,  what  is  the  area? 

2.  Find  the  area  of  a  piston  whose  diameter  is  5  in. 

3.  Find  the  diameter  of  a  piston  whose  area  is  19.635 
sq.  in. 

4.  What  is  the  diameter  of  a  piston  if  its  area  is  15.9 
sq.  in.  ? 

5.  The  total  force  F  pressing  against  the  end  of  a  piston 
equals  the  product  of  the  area  in  square  inches  by  the  pres- 
sure per  square  inch.  This  may  be  expressed  by  the  for- 
mula F  =  Pa,  in  which  F  is  the  total  number  of  pounds; 

also  by  the  formula 

„       P  3.1416  ^2  p7r^2 

F  =  -. or  F  =  —r—' 

4  4 

Explain  how  this  formula  is  obtained. 

6.  If  the  pressure  in  a  cylinder  is  50  lbs.  per  sq.  in.  and 
the  area  of  the  piston  is  16.4  sq.  in.,  what  is  the  total  force  (F) 
pressing  against  the  piston? 

7.  If  the  pressure  (P)  is  55  lbs.  per  sq.  in.  and  the  diam- 
eter of  the  piston  5  in.,  what  is  the  total  force  (F)  pressing 
against  the  piston? 

8.  What  is  the  total  force  against  a  piston  during  the 
power  or  explosion  stroke  of  an  engine,  if  the  diameter  of 
the  cylinder  is  4.5  in.  and  the  average  pressure  is  65  lbs. 
per  sq.  in.  ? 

9.  (a)  Find  the  area  of  the  piston  if  a  pressure  of  60  lbs. 
per  sq.  in.  produces  a  total  pressure  of  756  lbs. 

(b)     Find  the  diameter  of  the  piston. 

10.  A  pressure  of  80  lbs.  per  sq.  in.  on  a  piston  produces 
a  total  force  of  570r8  lbs.  What  is  the  diameter  of  the 
piston  ? 


AUTOMOBILE  ENGINES  9 

The  total  amount  of  energy  generated  during  one  explosion 
stroke  may  be  expressed  in  inch-pounds;  820  in.-lbs.  will  lift 
820  lbs.  through  1  in.  or  410  lbs.  through  2  in.,  etc. 

Describe  other  illustrations  of  work  done  by  inch-pounds. 
The  total  amount  of  energy  {E)  is  the  product  of  the  total 
force  (F)  pressing  against  the  piston  and  the  length  (Z) 
through  which  the  piston  acts.  If,  during  one  explosion 
stroke,  a  total  average  force  of  2500  lbs.  is  exerted  through 
a  distance  of  6  in.,  the  total  amount  of  energy  generated  is 
15,000  in.-lbs.,  since  2500  X  6  =  15,000. 

11.  (a)  How  many  inch-pounds  are  developed  if 
F  =  1900  and  /  =  8  in.  ? 

(b)  Explain  the  formula  E  =  Fl. 

(c)  Since  F  =  Po,  explain  how  the  formula  E  =  Fl 
becomes  E  =  Pla. 

(d)  900  in.-lbs.  lifts  900  lbs.  through  1  In.,  or  through  j\ 
of  a  foot;  hence  900  in.-lbs.  =  75  ft.-lbs.     Why.? 

(e)  Explam  the  formula  E  =  —  =  jy^    =  -^g-,  m 

which  E  denotes  the  number  of  foot-pounds  of  energy  de- 
veloped during  one  explosion  stroke. 

12.  How  many  foot-pounds  of  energy  are  generated  by 
one  explosion  in  an  engine  if  the  diameter  of  the  cylinder  is 
4  in.,  the  length  of  the  stroke  (Z)  is  5  in.,  and  the  average 
pressure  (P)  is  62.5  lbs.  per  sq.  in.  ? 

13.  If  d  is  5.5  in.,  I  is  4.6  in.,  P  is  65  lbs.  per  sq.  in.,  how 
many  foot-pounds  of  energy  are  generated  during  each  ex- 
plosion ? 

14.  If  the  area  (a)  of  the  piston  of  an  engine  is  18  sq.  in., 
the  length  of  the  stroke  5  in.,  and  the  energy  developed  at 
each  explosion  435  ft.-lbs.,  what  is  the  mean  average  pres- 
sure (P)  ? 


10 


APPLICATIONS  OF  ALGEBRA 


Find  the  value  of  the  unknown  in  each  of  the  following 
problems : 


DIAM. 

ENERGY 

PRESSURE 

LENGTH 

OF 
PISTON 

(during  one 
explosion) 

Foot-Pounds 

15 

50  lbs.  per  sq.  in. 

4     in. 

4.5  in. 

X 

16 

X  "       "    "     " 

5     " 

4     " 

251.328 

17 

60  "       "    "     " 

X        " 

4     " 

345.576 

18 

50  "       "    "     " 

5     " 

4.8  " 

X 

19 

48  "       "    «     « 

4.8  " 

X      " 

376.992 

6.    Formula  for  Energy  per  Minute. 
If  n  =  the  number  of  revolutions  of  the  crank-shaft  per 
minute,  then  n  =  the  number  of  revolutions  of  the  fly- 


wheel per  minute.     Why? 


And  ^  =  the  number  of  ex- 


plosions in  a  single  cylinder  per  minute.     Why  ? 

Since  E  =  the  number  of  foot-pounds  of  energy  developed 

during  one  explosion  stroke  and  E  =  -j^y  then 

the  number  of  foot-pounds  of  energy  developed  in  a  single 
cylinder  in  one  minute.     Why  ? 


12   '^  2 


Fn^^.nr.      Ene^'gy  Pe^  minute,   -p  _  Pla      n  _  Plan 
Formula.  ^^^^  ^^^^^^^        ^  '  1^  ^  2  '  -2^' 


Illustrative  Problem 

A  one-cylinder  gas-engine  makes  420  explosions  per  min- 
ute, the  inside  diameter  of  the  cyhnder  is  4  in.,  the  length  of 
the  stroke  is  5  in.  Find  the  pressure  per  square  inch  of  the 
piston  if  the  engine  develops  109,956  ft. -lbs.  of  energy  per 
minute. 


AUTOMOBILE  ENGINES 


11 


E  =   109,956  ft.-lbs.  per  minute, 
I   =   5  in. 

4 
n    =  2     X    420    =   840. 
Plan 


a    = 


47r  =  12.5664  sq.  in.     Why? 


Then  the  formula  E  = 


becomes        109,956  = 


24 

PX  5X12.5664X840 
24 


or 


2199. 12P  =  109,956  and  P  =  50  lbs.  per  sq.  in. 


7.  Problems  on  Revolutions  of  the  Fly- Wheel.  In  the 
following  problems  n  denotes  the  number  of  revolutions  per 
minute  of  the  fly-wheel,  and  E  the  number  of  foot-pounds 
developed  in  one  cylinder  per  minute. 

1.  If  the  fly-wheel  of  a  one-cylinder  engine  makes  400 
revolutions  per  minute,  the  pressure  (P)  is  64  lbs.  per  sq.  in., 
the  length  (l)  is  4  in.,  and  the  area  (a)  is  16  sq.  in.,  how  many 
foot-pounds  of  energy  are  developed  per  minute  ? 

2.  The  fly-wheel  of  a  gas-engine  makes  960  revolutions 
per  minute,  P  is  70  lbs.  per  sq.  in.,  and  I  is  4.5  in.  What  is 
the  area  of  the  piston  if  158,400  ft.-lbs.  of  energy  are  de- 
veloped per  minute  in  a  single  cylinder  ? 

Find  the  value  of  the  unknown  in  each  of  the  following 
problems,  if  E  denotes  the  energy  developed  per  minute  in 
a  single  cylinder. 


P 

I 

a 

n 

E 

3 

66 

5.6 

14 

1000 

'     X 

4 

X 

5.5 

15 

800 

176,000 

5 

80 

X 

13.5 

900 

182,250 

6 

75 

5 

X 

1200 

243,750 

7 

70 

5.2 

18 

X 

300,300 

8 

72 

4.4 

20.5 

1000 

X 

9 

X 

4.8 

19.2 

1500 

345,600 

12 


APPLICATIONS  OF  ALGEBRA 


If  c  equals  the  number  of  cylinders  in  a  gas-engine  the 

number  of  foot-pounds  of  energy  {E)  developed  per  minute 

•  •       •  Plo/ftc 

m  the  gas-engine  is  expressed  by  the  formula  E  =■  . 

ATX. 

Explain. 

10.  A  gas-engine  has  4  cylinders,  each  having  a  diameter 
of  4  in.  If  the  length  of  the  stroke  is  5  in.,  the  average 
pressure  on  the  piston  is  60  lbs.  per  sq.  in.,  and  the  number 
of  revolutions  of  the  fly-wheel  per  minute  is  600,  how  many 
foot-pounds  of  energy  are  developed  per  minute  ? 

11.  A  two-cylinder  gas-engine  develops  360,000  ft. -lbs.  of 
energy  per  minute,  A  is  16  sq.  in.,  I  is  4.5  in.^  and  n  is  800 
revolutions  per  minute.  What  is  the  average  pressure  (P) 
per  square  inch  against  each  piston  ? 

If  E  denotes  the  energy  developed  in  the  gas-engine,  find 
X  in  the  following  problems: 


P 

/ 

a 

n 

c 

E 

12 

65 

4.2 

18 

1000 

4 

X 

13 

70 

X 

15.5 

1400 

4 

1,063,300 

14 

72 

4.8 

X 

850 

4 

628,320 

15 

69 

5 

16 

1300 

X 

1,196,000 

16 

74 

5.3 

17.5 

900 

8 

X 

8.    Problems  on  Horse-Power.    Let  HP  equal  the  horse- 
power of  a  gas-engine.     Since  33,000  ft.-lbs.  per  minute  is 
1  horse-power,  the  horse-power  of  the  engine  is  the  number 
of  foot-pounds  developed  per  minute  divided  by  33,000. 
If  P  =  pressure  in  pounds  per  square  inch, 
I  —  length  of  stroke  in  inches, 
a  =  area  of  piston  in  square  inches, 
c  =  number  cylinders  in  engine, 
E  =  energy  during  one  explosion  stroke  in  foot-pounds. 


AUTOMOBILE  ENGINES 


13 


n  =  number  of  revolutions  of  the  fly-wheel  per  minute, 
__      E      _        Plane        _  PlT(Pnc 

then  HF  -  ^^-q^  -  24  X  33,000  ~  24  X  4  X  33,000 

1.  How  many  horse-power  are  developed  by  a  four- 
cylinder  engine  if  P  is  60  lbs.  per  sq.  in.,  Z  is  5  in.,  a  is  16.5 
sq.  in.,  and  n  is  1000  revolutions  per  minute? 

2.  A  gas-engine  has  four  cylinders.  What  is  the  aver- 
age pressure  (P)  per  square  inch  if  I  is  4.8  in.,  a  is  17.6  sq.  in., 
n  is  1350  revolutions  per  minute,  and  the  horse-power  is  48  ? 

3.  How  many  revolutions  per  minute  will  the  fly-wheel 
of  a  four-cylinder  gas-engine  make  while  developing  40.5 
horse-power,  if  P  is  66  lbs.,  I  is  4.5  in.,  and  a  is  20  sq.  in.  ? 

Find  X  in  the  following  problems: 


P 

/ 

d 

n 

c 

HP 

4 

72 

5.5 

4 

1250 

4 

X 

5 

80 

5.5 

5 

X 

6 

65.45 

6 

X 

5 

4 

1400 

4 

31 

7 

77 

X 

4.8 

1360 

4 

50.26 

8 

71 

5.5 

X 

1000 

4 

40.3 

9 

68.5 

4.8 

5 

1450 

4 

X 

10 

72 

5 

4 

1500 

X 

68.5 

The  California  State  Automobile  Registration  Board  dur- 
ing 1915  used  the  formula  HP  =  .224((f  -f-  l)dc.  By  this 
formula  find  the  horse-power  in  each  of  the  problems  4,  5, 
6,  and  9  above.     The  formula  used  by  the  State  Board  in 


1916  is  HP  =  ^  d?c 
5 


By  this  formula  find  the  horse-power 
in  problems  4,  5,  6,  and  9. 


CHAPTER  II 


DRIVING 
GEAR 


TRANSMISSION  OF  SPEED 

9.    Gears   in   Mesh.    The   accompanying  figure   shows 
two  gears  in  mesh,  the  smaller  gear,  or  pinion  (A),  turns  the 
driven  gear  (B).     It  is  not  difficult  to  see  that  the  driving- 
gear  rotates  in  one  direction 
while  the  driven  gear  rotates 
in  the  opposite  direction. 

(a)  Count  the  number  of 
teeth  in  each  gear. 

(b)  When  the  driving- 
gear  (A)  turns  once,  with  how 
many  teeth  of  the  driven  gear 
{B)  has  it  been  in  mesh  ? 

(c)  How  many  times  must 
the  point  c  on  the  driving- 
gear  return  to  its  position  in 
order  to  put  each  tooth  of  B 

in  mesh  once,  or  to  cause  one  complete  revolution  oi  B? 
Let  t  represent  the  number  of  teeth  in  A, 
t'  represent  the  number  of  teeth  in  B, 
r  represent  the  number  of  revolutions  in  Ay 
r'  represent  the  number  of  revolutions  in  B, 
Show  by  counting  that  the  following  numerical  ratios 
are  true: 


DRIVEN 
GEAR 


1.     7/  =  ^  =  o*     '^^^^  equation  is  read: 


'The  number 


of  teeth  in  A  is  to  the  number  of  teeth  in  5  cw  16  is  to  32, 
or  as  1  to  2." 

14 


TRANSMISSION  OF  SPEED  15 

2.  Show  by  counting  the  revolutions  that 

r      2       ,  r'       1 
-,=  jand-=-. 

t         1  r'        1  t        r'     . 

3.  Since  r,  =  ?:  and  -  =  i^,  then  -,  =  — ,  since  the  value 

r       2         r        2  t'       r 

t  r'   .    1 

of  each  ratio,  7?  and  — ,  is  ».     This  equation  may  be  ex- 

Z  T  JL 

pressed  by  saying  that  the  number  of  teeth  in  two  gears 
in  mesh  is  inversely  proportional  to  the  number  of  revolu- 
tions they  make  in  a  given  time. 

t        r' 
By  clearing  of  fractions  the  equation  p  =^  —  becomes 

tr  =  tV. 

In  the  accompanying  figure  <  =  16,  r  =  2,  /'  =  32, 
r'  =  1;  the  equation  tr  =  t'r'  becomes  16  X  2  =  32  X  1. 

This  equation  may  be  expressed  by  saying  that  the  number 
of  revolutions  of  the  driving-gear  multiplied  by  the  num- 
ber of  its  teeth  equals  the  number  of  revolutions  of  the  driven 
gear  multiplied  by  the  number  of  its  teeth. 


10.    Problems  on  Gears. 

1.  A  driving-gear  with  12  teeth  makes  200  revolutions 
per  minute.  How  many  revolutions  per  minute  will  be 
made  by  the  driven  gear  which  has  24  teeth  ? 

2.  A  driving-gear  with  20  teeth  revolves  180  times  per 
minute.  How  many  revolutions  per  minute  will  the  driven 
gear  make  if  the  number  of  its  teeth  is  30?  How  many  if 
the  number  of  its  teeth  is  24  ? 

3.  A  gear  having  16  teeth  is  making  240  revolutions  per 
minute.  How  many  teeth  has  a  second  gear  in  mesh  with 
the  first  if  it  is  making  120  revolutions  per  minute?  How 
many  teeth  would  the  second  gear  have  if  it  were  making 
160  revolutions  per  minute? 


16  APPLICATIONS  OF  ALGEBRA 

Complete  the  following  table  by  filling  in  the  blank  spaces. 


DRIVING-GEAR 

DRIVEN  GEAR 

NO.  OF 

NO.  OF 

NO.  OF 

NO.  OF 

REV. 

TEETH 

REV. 

TEETH 

4 

180 

16 

20 

5 

360 

12 

540 

6 

750 

,  . 

600 

25 

7 

, , . 

22 

462 

28 

8 

384 

27 

32 

9 

155 

85 

31 

10 

620 

13 

260 

11 

. . . 

24 

288 

31 

12 

363 

15 

. . . 

55 

13 

275 

14 

78 

•• 

11.  High-Speed  Transmission.  The  motion  from  the 
engine  which  turns  the  crank-shaft  iC.S.)  and  fly-wheel 
(F.  W.)  reaches  the  rear  axle  by  means  of  the  main  trans- 
mission-shaft, T. 

The  front  part  of  the  main  transmission-shaft  is  connected 
with  the  engine  crank-shaft  by  means  of  the  cone  clutch 
(C.C.),  the  two  conical  surfaces  of  which  are  held  firmly  to- 
gether by  springs,  except  when  forced  apart  by  the  driver 
pressing  the  clutch  pedal,  as  in  the  figure  showing  low  speed 
(page  30) .  With  this  exception  the  front  part  of  the  transmis- 
sion-shaft (T)  revolves  whenever  the  engine  runs.  The  rear 
part  (S)  communicates  its  motion  through  the  bevel  pinion 
and  gear  (Z  and  J)  to  the  rear  axle  and  the  rear  wheels. 

The  accompanying  figure  shows  the  high-speed  transmis- 
sion which  gives  the  greatest  possible  car  speed  for  a  given 
speed  of  the  engine.  This  manner  of  connection  is  also  called 
the  direct  drive  (D.D.), 

One  part  of  the  clutch  DD  is  fixed  to  the  gear  G.  When 
this  is  brought  into  its  extreme  forward  position,  as  shown 
in  the  figure,  the  clutch  DD  engages  and  T  and  S  rotate  as 


17 


18  APPLICATIONS  OF  ALGEBRA 

one  shaft.  This  manner  of  connection  is  called  the  direct 
drive.  It  is  also  called  high-speed  transmission,  since  it  gives 
the  greatest  possible  car  speed  for  a  given  speed  of  the  engine. 
The  direct-drive  transmission  does  not  use  the  counter-shaft 
R,  but  because  gear  F  is  in  constant  mesh  with  gear  A  the 
counter-shaft  turns  ready  for  use  when  intermediate,  low,  or 
reverse  speeds  are  desired. 

Without  changing  the  position  of  the  gears  the  speed  of 
the  car  may  be  regulated  to  some  extent  by  increasing  or 
decreasing  the  amount  of  gasolene  used  in  the  engine,  or  by 
advancing  or  retarding  the  spark. 

The  following  questions  refer  to  the  direct-drive  position 
of  the  gears. 

1.  In  this  position  which  gears  and  shafts  are  rotating 
as  the  engine  runs?  Which  are  idling  {i.  e.,  turning  without 
transmitting  power)  ? 

2.  What  is  the  effect  of  disengaging  the  cone  clutch  ? 

3.  What  is  the  effect  of  stopping  the  explosions  in  the 
engine  ?     What  parts  will  continue  to  turn  ? 

4.  What  is  the  effect  of  disengaging  D.D.  when  the  engine 
is  running  and  C.C.  is  engaged.'* 

5.  With  both  clutches  engaged,  one  turn  of  the  crank- 
shaft makes  how  many  turns  of  the  pinion  /  ? 

12.  Problems  on  High-Speed  Transmission.  In  each  of 
the  following  problems  which  concern  four-cylinder  engines 
the  two  clutches  are  connected  to  form  the  direct  drive. 

1.  If  a  gas-engine  makes  920  explosions  per  minute,  how 
many  times  will  the  fly-wheel  revolve  per  minute  ? 

2.  If  the  clutch  is  thrown  in,  how  many  times  will  the 
small  pinion-gear  at  the  rear  of  the  main  driving-shaft  re- 
volve per  minute? 


TRANSMISSION  OF  SPEED 


19 


3.  If  the  driving-pinion  contains  13  teeth  and  is  in  mesh 
with  a  driven  bevel-gear  which  contains  52  teeth,  how  many 
times  will  the  driven  gear  turn  when  the  driving-pinion 
turns  once  ?  when  the  driving-pinion  turns  460  times  ? 

4.  If  the  fly-wheel  makes  920  revolutions  per  minute, 
how  many  times  will  the  rear  axle  turn  if  the  bevel-gear 
ratio  is  1  to  4  ?  How  far  will  the  car  travel  in  a  minute  if 
the  circumference  of  the  rear  wheel  is  9  ft.  ? 

Complete  the  following  table  in  problems  5  to  15  by  filling 
in  the  blanks.  Consider  the  bevel-pinion  as  having  13  teeth, 
the  bevel-gear  as  having  52  teeth,  and  the  circumference  of 
the  rear  wheel  as  9  ft. 


NO.  REV. 

NO.  REV. 

NO.  REV. 

NO.  REV. 

NO.  FEET 

OF  FLY- 

OF DRIV- 

OF BEVEL- 

OF   REAR 

TRAVELLED 

WHEEL 

ING-PINION 

GEAR 

AXLE 

PER  MIN. 

5 

900 

.... 

6 

1000 

.  .  . 

7 

1200 

,  .  , 

8 

1280 

.  .  . 

9 

840 

10 

325 

11 

. . . 

275 

12 

310 

13 

290 

14 

.  .  . 

.  .  . 

2385 

15 

2655 

16.  If  the  speed  limit  in  the  country  is  25  miles  per  hour, 
which  of  the  preceding  problems  have  rates  which  exceed 
the  speed  limit  ? 

13.  Gear-Box  and  Gears  in  Neutral.  The  purpose  of 
the  gear-box  mechanism  (G.B.)  is  to  enable  the  two  parts 
T  and  S  of  the  transmission-shaft  to  be  connected  in  either 
of  four  ways  or  to  be  entirely  disconnected,  the  selection 
being  made  by  the  driver  placing  the  hand-lever  in  one  of 


20  APPLICATIONS  OF  ALGEBRA 

five  positions.  When  disconnected  the  lever  and  gear  mech- 
anism are  said  to  be  neutral,  as  in  the  figure.  Show  why 
shaft  S  does  not  rotate  when  T  rotates.  (Study  D.D.)  The 
four  methods  of  connection  are  called  first  or  low  speed, 
second  or  intermediate  speed,  third  or  high  speed,  and  reverse 
speed.  The  words  low  and  high  do  not  refer  to  the  actual 
running-speed  of  the  car,  but  to  the  ratio  of  the  speed  of  the 
car  to  that  of  the  engine,  which  is  determined  by  the  speed 
ratio  of  the  two  parts  T  and  *S  of  the  transmission-shaft. 

The  front  part  T  of  the  transmission-shaft  carries  the 
gear  F  and  one  part  of  the  clutch  D.D.  The  other  part  of 
this  clutch  is  fixed  to  the  gear  G.  This  clutch  D.D.  is  en- 
gaged only  in  the  direct-drive  or  high-speed  transmission 
when  T  and  S  rotate  as  one  shaft.  Gears  G  and  H  are  car- 
ried upon  the  square  part  of  shaft  S  so  as  to  rotate  with  it, 
but  may  slide  along  it.  It  is  by  sliding  G  and  H  into  dif- 
ferent positions  that  the  manner  of  connection  of  <S  and  T 
is  determined.  The  gears  A,  B,  C,  and  D  are  fixed  to  the 
counter-shaft  R. 

In  the  preceding  figure,  the  direct-drive  clutch  (D.D.)  is 
disengaged,  S  and  T  being  entirely  disconnected.  This  is 
usually  the  case  when  the  engine  is  running  idle,  i.  e.,  without 
driving  the  car.  (The  engine  may  be  permitted  to  run  idle 
by  disengaging  the  cone  clutch,  whatever  the  connection  of 
S  and  T  may  be,  but  this  requires  that  the  clutch  pedal  be 
forcibly  held  down  by  the  driver.)  This  position  is  shown 
in  the  following  figure  for  low-speed  transmission.  The  car 
here  cannot  run  until  the  driver  releases  the  cone  clutch. 

1.     Locate  the  following  parts  in  the  accompanying  figure. 

F.  IF.— Fly-wheel. 

C.C. — Cone  clutch. 

T  and  S — Main  transmission-shaft,  front  and  rear  parts. 

D.D. — Direct-drive  clutch. 

G.B. — Gear-box. 


i!^Z^"lII 


T 


22  APPLICATIONS  OF  ALGEBRA 

I — Bevel-pinion. 

J — Bevel-gear. 

R.A. — Rear  axle. 

R — Counter-shaft. 

A  and  F — Constant  mesh  gears. 

B  and  G — Intermediate-speed  gears. 

C  and  H — Low-speed  gears. 

D  and  E — ^Reverse-speed  gears  (with  H). 

2.  Which  parts  rotate  when  the  cone  clutch  C.C.  is  dis- 
engaged ? 

3 .  Which  parts  rotate  when  the  cone  clutch  C.  C  is  engaged 
and  the  direct-drive  clutch  D.D.  disengaged  and  the  gears  are 
located  as  in  the  preceding  figure? 

4.  When  C.C.  is  engaged  and  D.D.  disengaged,  which 
parts  rotate: 

(a)  When  B  and  G  are  in  mesh,  as  in  figure  showing 
intermediate  speed? 

(b)  When  C  and  H  are  in  mesh,  as  in  figure  showing 
low  speed? 

(c)  When  E  and  H  are  in  mesh,  as  in  figure  showing 
reverse  speed? 

14.  Intermediate  Speed.  To  run  an  engine  at  an  inter- 
mediate speed  a  counter-shaft  {R)  is  used.  On  this  shaft 
are  four  gear-wheels  {A,  B,  C,  and  D)  rigidly  attached.  On 
part  (T)  of  the  main  transmission-shaft  the  gear-wheel  F  is 
firmly  fixed  and  in  constant  mesh  with  the  gear-wheel  A  on 
the  counter-shaft  R.  When  T  rotates,  the  shaft  R  and  all  of 
the  gear-wheels  on  it  rotate  also. 

The  section  of  the  transmission-shaft  S  which  is  within 
the  gear-box  is  square,  except  the  bearings,  one  of  which  is 
within  the  hub  of  the  gear  F.  The  two  gears  G  and  H  have 
square  holes  in  their  hubs  that  fit  the  square  shaft  S;  thus 
when  these  gears  revolve  the  shaft  also  revolves. 

For  the  intermediate  speed,  the  gear  G  is  moved  along  S 
until  it  engages  with  B.     The  clutch  D.D.  is  now  "out,"  so 


23 


24  APPLICATIONS  OF  ALGEBRA 

that  T  and  S  are  not  connected  directly,  but  the  motion  of 
the  fly-wheel  is  transmitted  through  the  gears  F,  A,  B,  G, 
Z,  J  to  the  rear  wheel.  Trace  this  transmission  in  the  figure 
above. 

Use  the  figure  illustrating  intermediate  speed  in  answer- 
ing the  following  questions.  The  gears  in  the  figure  have 
the  following  numbers  of  teeth: 

F,  15  J5,  28  I,  13 

A,  30  G,  28  J,  52 

(1)  When  F  turns  twice,  how  many  times  does  A  turn? 
Why? 

(2)  In  this  figure  why  does  not  G  turn  the  same  number 
of  times  as  F? 

(3)  Why  does  gear  B  turn  the  same  number  of  times  as 
gear  A  ? 

(4)  Why  does  gear  G  turn  the  same  number  of  times  as 
gear  B  ? 

(5)  Why  does  pinion  I  turn  the  same  number  of  times  as 
gear  G  ? 

(6)  Why  does  pinion  I  turn  the  same  number  of  times 
as  gear  B  ? 

(7)  Why  does  pinion  I  turn  the  same  number  of  times 
as  gear  A  ? 

(8)  Why  does  pinion  I  turn  one-half  as  often  as  gear  F  ? 

(9)  Why  does  pinion  I  turn  one-half  as  often  as  the 
fly-wheel  ? 

(10)  In  considering  gears  F  and  Ay  why  is  F  the  driving- 
gear  and  A  the  driven  gear  ?  In  considering  gears  B  and  G 
which  is  the  driving-gear,  B  or  G? 

Illustrative  Problems 

Let  F,  the  first  driving-gear,  contain  17  teeth, 
and  Ay  the  first  driven  gear,  contain  31  teeth, 
and  By  the  second  driving-gear,  contain  24  teeth, 
and  G,  the  second  driven  gear,  contain  31  teeth. 


TRANSMISSION  OF  SPEED  25 

Then  if  the  fly-wheel  turns  1800  times,  how  many  times 
does  pinion  I  turn? 

Justify  each  of  the  following  statements: 

When  the  fly-wheel  turns  once,  gear  F  turns  once. 

When  the  fly-wheel  turns  1800  times,  F  turns  1800  times. 

When  gear  F  turns  once,  gear  A  turns  ^  times. 

When  gear  F  turns  1800  times,  gear  A  turns  1800  X  if 

times. 

When  gear  A  turns  once,  gear  B  turns  once.     (See  figure.) 
When  gear  A  turns  1800  X  if  times,  gear  B  turns  1800 

X  H  times.     Why  ? 
When  gear  B  turns  once,  gear  G  turns  ff  times. 
When  gear  B  turns  1800  X  if  times,  gear  G  turns  1800 

X  if  X  M  times. 
So,  when  the  fly-wheel  turns  1800  times,  F  turns  1800 

times,  gears  A  and  B  each  turn  1800  X  if  times,  and  gear 

G  and  pinion  I  turn  each  1800  X  if  X  If  times. 
The  number   1800  X  if  X  |f  may  be  obtained  directly 

by  noticing  its  factors. 

1800  is  the  number  of  revolutions  of  the  fly-wheel, 
17  is  the  number  of  teeth  of  the  first  driving-gear  F, 
24  is  the  number  of  teeth  of  the  second  driving-gear  J5, 
31  is  the  number  of  teeth  of  the  first  driven  gear  A, 
31  is  the  number  of  teeth  of  the  second  driven  gear  G. 

Hence  to  find  how  many  times  the  pinion  turns,  find  the 
product  of  the  number  of  revolutions  of  the  fly-wheel,  the 
number  of  teeth  on  the  first  driving-gear,  and  the  number  of 
teeth  on  the  second  driving-gear,  and  divide  by  the  product 
of  the  number  of  teeth  on  the  first  driven  gear  and  the  num- 
ber of  teeth  on  the  second  driven  gear. 

The  same  principle  may  be  stated  as  a  formula: 
Let  r    =  number  of  revolutions  of  the  fly-wheel. 
Let  ti   =  number  of  teeth  in  the  first  driving-gear, 
(read  t  one) 


26 


APPLICATIONS  OF  ALGEBRA 


Let  U   =  number  of  teeth  in  the  second  driving-gear. 

(read  t  two) 
Let  ti    =  number  of  teeth  in  the  first  driven  gear. 

(read  t  one  prime) 
Let  U'  =  number  of  teeth  in  the  second  driven  gear. 

(read  t  two  prime) 

Then  the  number  of  revolutions  of  the  pinion  is    /   ,. 

(a)     Show  how  each  factor  of  the  number  1800  X  H  X 

If  is  represented  in  the  number  tttI' 

h  h 

In  the  same  way,  if  there  are  three  driving-gears  and  three 
driven  gears,  the  number  of  revolutions  of  the  third  driven 
gear  is  r  ti  <2  U/ti  i^  U'. 

15.    Problems  on  Intermediate  Speed. 

1.  If  the  fly-wheel  turns  1600  times,  use  the  preceding 
formula  to  find  the  revolutions  of  the  pinion  when  the  gears 
have  the  following  numbers  of  teeth: 


F,  15 
^,30 


5,28 
G,  28 


7,13 
J,  52 


Use  the  formula  to  complete  the  following  table: 


FLY- 
WHEEL 

DRIVING- 
GEARS 

DRIVEN 
GEARS 

REAR 
WHEEL 

NO.  OF 
REV. 

NO.  OF 
TEETH 

NO.  OF 
TEETH 

NO.  OF 
REV. 

F 

B 

I 

A 

G 

J 

2 
3 

4 
5 
6 

7 
8 

10571 

iosn 

21142 

528 

5500 

17 
17 
17 
17 
18 
18 
18 

24 
24 

13 

28 

18 

15 
15 
15 
15 
15 
15 
15 

31 
31 
31 
31 
36 
36 
36 

31 
31 
31 

28 
36 
36 

55 
55 
55 
55 
55 
55 
55 

2448 

867 
1326 

375 

72 

TRANSMISSION^  OF  SPEED  27 

9.  (a)  How  many  times  will  the  pinion  I  turn  while 
the  fly-wheel  makes  40  revolutions  when  the  number  of 
teeth    is    as    follows: 


h  =  15 

fe  =  28 

<3  =  13 

</  =  30 

^'=28 

h'  =  52 

(b)  Show  from  the  figure  why  pinion  /  is  named  tz  and 
gear  J  is  named  U', 

(c)  When  U  turns  once,  how  many  times  does  tz  turn  ? 

(d)  How  many  times  does  the  bevel-gear  turn  when  the 
pinion  turns  20  times? 

(e)  How  many  times  does  the  rear  wheel  turn  when  the 
fly-wheel  turns  40  times? 

(f)  Explain  the  following  from  the  figure: 

When  the  fly-wheel  turns  once  the  pinion  turns  \  time. 

When  the  pinion  turns  once  the  bevel-gear  turns  \  time. 

When  the  pinion  turns  \  time  the  bevel-gear  turns  i  X  i 
time. 

When  the  fly-wheel  turns  once  the  bevel-gear  turns  \ 
time. 

So  when  the  fly-wheel  turns  once  the  rear  wheel  turns  ^ 
time. 

Trace  from  the  figure  the  development  of  the  following 
ratio,  which  is  called  Ihe  final-gear  ratio. 

number  of  revolutions  of  the  rear  wheel  _  1 
number  of  revolutions  of  the  fly-wheel       8* 

10.  (a)  When  the  fly-wheel  turns  1200  times  a  minute, 
how  many  revolutions  does  the  rear  wheel  make  ? 

If     n  =  the  number  of  revolutions  of  the  fly-wheel  per 
minute, 
r  =  final  gear  ratio, 
then    nr  =  the  number  of  revolutions  of  the  rear  wheel. 
Illustrate, 
(b)     When  the  fly-wheel  turns  1200  times  a  minute  and 


28  APPLICATIONS  OF  ALGEBRA 

the  circumference  of  the  rear  wheel  is  9  ft.,  how  far  does  the 
machine  travel  in  a  minute  ? 

If        nr  =   the  number  of  revolutions  of  the  rear  wheel 
and       c  =   the  circumference  of  the  rear  wheel  in  feet, 
then  nrc  =  the  number  of  feet  travelled  by  the  machine 
in  a  minute.     Explain. 

(c)  When  the  fly-wheel  turns  1200  times  a  minute  and 
the  circumference  of  the  rear  wheel  is  9  ft.,  how  far  does  the 
machine  travel  in  25  minutes  on  the  intermediate  speed  ? 

li   d  =  the  distance  in  feet, 
and  m  =  the  number  of  minutes  travelled, 
explain  the  formula,    d  =  nrcm. 

The  total  number  of  feet  travelled  by  an  automobile  is  ex- 
pressed by  the  product  of  the  number  of  revolutions  of  the  fly- 
wheel per  minute,  the  given  gear  ratio,  the  circumference  of 
the  rear  wheel  in  feet,  and  the  number  of  minutes  travelled. 

In  the  following  set  of  problems  the  number  of  teeth  on 
the  gears  are  the  same  as  those  given  in  9(a),  preceding, 
which  make  the  final  gear  ratio  ^.  The  gears  are  set  at 
intermediate  speed. 

11.  If  the  fly-wheel  of  an  automobile  makes  840  revolu- 
tions per  minute  and  the  circumference  of  the  rear  wheel  is 
8  ft.,  how  many  feet  will  the  car  travel  in  10  minutes? 

12.  The  fly-wheel  makes  960  revolutions  per  minute, 
and  the  circumference  of  the  rear  wheel  is  8^  ft.  How  far 
will  the  car  travel  in  5  minutes? 

13.  How  many  revolutions  per  minute  will  the  fly-wheel 
make  if  the  car  travels  one  mile  in  6  minutes  and  the  cir- 
cumference of  the  rear  wheel  is  8  ft.  ? 

14.  A  motor-car  whose  rear  wheel  has  a  circumference 
of  8i  ft.  travels  2  miles  in  10  minutes.  How  many  revo- 
lutions per  minute  does  the  fly-wheel  make  ? 


TRANSMISSION  OF  SPEED  29 

15.  How  many  minutes  will  it  take  an  auto  to  travel 
2  miles  if  the  circumference  of  the  rear  wheel  is  8  ft.  and 
the  fly-wheel  makes  1056  revolutions  per  minute? 

16.  How  long  will  it  take  a  car  to  run  3  miles  if  the  fly- 
wheel makes  880  revolutions  per  minute  and  the  circum- 
ference of  the  rear  wheel  is  8  ft.  ? 

17.  What  must  the  circumference  of  the  rear  wheel  be 
if  the  fly-wheel  makes  880  revolutions  per  minute  and  the 
car  goes  3400  ft.  in  4  minutes  ? 

18.  A  car  travelled  3^  miles  in  16  minutes  while  the  fly- 
wheel made  1140  revolutions  per  minute.  What  was  the 
circumference  of  the  rear  wheel  ? 

16.  Problems  on  Low  Speed.  To  set  the  gears  for  low 
speed,  the  driver,  by  means  of  the  hand-lever,  slides  gear 
H  into  mesh  with  C.  Gear  G  and  clutch  D,  D.  are  now  in  the 
same  position  as  in  neutral,  both  being  disconnected.  Thus 
the  motion  of  the  fly-wheel  is  transmitted  through  the  gears 
F,  A,  C,  H^  7,  and  J  to  the  rear  wheel.  Trace  this  trans- 
mission in  the  figure.  Notice  that  the  cone  clutch  is  open 
and  must  be  closed  before  the  car  will  move. 

Answer  the  following  questions,  the  gears  being  set  at  low 
speed: 

1.  If  F  has  15  teeth,  A  30,  C  15,  H  30,  I  13,  and  J  52, 
how  many  revolutions  will  F  make  while  the  fly-wheel 
makes  80?  How  many  will  A  make?  C?  H?  I?  J? 
How  many  revolutions  will  the  rear  wheel  make?  What 
is  the  ratio  of  the  number  of  revolutions  that  the  rear  wheel 
makes  to  the  number  that  the  fly-wheel  makes,  or  what  is 
the  final  gear  ratio  ? 

2.  In  the  following  problems  use  the  formula  d  =  nrcm 
and  the  low-speed  ratio  ^V-  Explain  how  this  ratio  is  ob- 
tained. 


30 


TRANSMISSION   OF  SPEED  31 

3.  How  many  revolutions  per  minute  will  the  fly-wheel 
make  if  the  circumference  of  the  rear  wheel  is  9^  ft.  and 
the  car  travels  5890  ft.  in  10  minutes  ? 

4.  If  an  auto  travels  2960  ft.  in  4  minutes  and  the  fly- 
wheel makes  1280  revolutions  per  minute,  what  is  the  cir- 
cumference of  the  rear  wheel  ? 

5.  In  how  many  minutes  will  a  car  run  a  mile  when  the 
fly-wheel  is  making  1100  revolutions  per  minute  and  the 
circumference  of  the  rear  wheel  is  8  ft.  ? 

6.  A  machine  is  delayed  by  a  flock  of  sheep  for  three- 
fourths  of  an  hour,  making  1^  miles.  How  many  times 
does  the  fly-wheel  turn  per  minute  if  the  circumference  of 
the  rear  wheel  is  8  ft.  ? 


17.  Reverse  Speed.  In  order  to  drive  the  car  back- 
ward, the  pinion  /  must  turn  in  the  opposite  direction  to 
that  which  causes  forward  motion  of  the  car.  This  re- 
quires that  S  and  T  must  be  so  connected  as  to  turn  in  op- 
posite directions.  This  is  accomplished  by  sliding  the  gear 
H  into  mesh  with  Ey  which  is  in  constant  mesh  with  Z). 
Gear  G  and  clutch  DD  remain  disengaged. 

(1)  If  the  fly-wheel  turns  counter-clockwise,  then  D 
turns  clockwise.     In  which  direction  does  pinion  I  turn? 

Trace  the  following  from  the  appropriate  drawings: 

(2)  During  low  speed,  if  the  fly-wheel  turns  counter- 
clockwise, then 

F  turns  counter-clockwise  on  main  transmission-shaft  T; 

A  turns  clockwise  (in  mesh  with  F) ; 

C  turns  clockwise  (on  counter-shaft  R) ; 

H  turns  counter-clockwise  (in  mesh  with  C) ; 

the  pinion  turns  counter-clockwise  (same  direction  as  H), 

and  the  rear  axle  turns  the  car  forward. 


32 


TRANSMISSION  OF  SPEED  33 

(3)     During  reverse  speed.    If  the  fly-wheel  turns  counter- 
clockwise, then 

F  turns  counter-clockwise  (on  main  transmission-shaft); 
A  turns  clockwise  (in  mesh  with  F) ; 
C  turns  clockwise  (on  counter-shaft  R) ; 
E  turns  counter-clockwise  (in  mesh  with  D); 
H  turns  clockwise  (in  mesh  with  E) ; 
Pinion  /  turns  clockwise  (same  direction  as  IT)\ 
and   the   rear   axle  turns  the   car   backward   (since,  when 
pinion  I  turned  counter-clockwise,  the  car  went  forward). 


CHAPTER  III 

AUTOMOBILES  IN  MOTION 
18.    Road  Problems. 

1.  If  F  has  15  teeth,  A  30,  D  12,  E  12,  H  30,  I  13,  and 
J  52,  what  is  the  final  reversing-gear  ratio  ? 

2.  How  long  will  it  take  a  car  to  back  180  ft.  with  the 
fly-wheel  turning  400  times  per  minute  if  the  final  gear  ratio 
is  -^  and  the  circumference  of  the  rear  wheel  is  9  ft.  ? 

3.  How  many  feet  per  minute  will  an  auto  make  on  the 
reverse  speed  if  the  fly-wheel  makes  500  revolutions  per 
minute,  the  gear  ratio  is  ^,  and  the  circumference  of  the  rear 
wheel  is  8  ft.  ?     What  will  be  the  rate  in  miles  per  hour  ? 

4.  A  man  wishing  to  find  the  reversing-gear  ratio  of  his 
car  sent  it  backward  85  ft.  in  20  seconds  while  the  fly-wheel 
was  turning  at  the  rate  of  480  times  per  minute.  What 
was  the  gear  ratio  if  the  circumference  of  the  rear  wheel  was 
8|ft.? 

5.  If  the  fly-wheel  of  a  car  revolves  1100  times  per 
minute,  its  gear  ratio  is  -j\,  and  the  circumference  of  the 
rear  wheel  is  9^  ft.,  how  many  feet  will  it  travel  in  5  minutes  ? 

6.  If  the  number  of  revolutions  of  the  fly-wheel  is  900, 
the  gear  ratio  is  t^,  and  the  auto  travels  5400  ft.  in  5  min- 
utes, what  is  the  circumference  of  the  rear  wheel  ? 

7.  How  many  minutes  will  it  take  an  automobile  to 
travel  5400  ft.  up  a  grade  if  the  fly-wheel  makes  880  revolu- 
tions per  minute,  the  low  gear  ratio  is  :f\,  and  the  circum- 
ference of  the  rear  wheel  is  9  ft.  ? 

34 


AUTOMOBILES  IN  MOTION  35 

8.  What  must  be  the  gear  ratio  of  a  car  if  it  travels 
12,825  ft.  in  5  minutes  when  the  number  of  revolutions  of 
the  fly-wheel  is  1045  per  minute  and  the  circumference  of 
the  rear  wheel  is  9  ft.  ? 

9.  How  many  hours  will  it  take  a  car  to  travel  from 
San  Francisco  to  Sacramento  (94  miles)  if  its  rate  is  1215 
ft.  per  minute? 

10.  How  many  hours  will  it  take  an  automobile  trav- 
eUing  1320  ft.  per  minute  to  go  from  Santa  Barbara,  Cali- 
fornia, to  Carson  City,  Nevada  (650  miles)  ? 

11.  How  many  feet  per  minute  must  a  motor-car  travel 
in  order  to  get  from  San  Diego  to  San  Bernardino  (153  miles) 
in  4|  hours.     What  will  its  rate  be  in  miles  per  hour  ? 

12.  What  is  the  distance  from  Berkeley  to  Yosemite 
Valley  if  an  automobile  travelling  at  the  rate  of  880  ft.  per 
minute  can  make  the  distance  in  22|^  hours  ? 

13.  If  the  fly-wheel  of  a  machine  makes  990  revolutions 
per  minute,  the  gear  ratio  is  ^,  the  distance  travelled  is  4 
miles  in  19f  minutes,  what  must  be  the  circumierence  of  the 
rear  wheel  ? 

14.  A  car  travels  from  Salem  to  Portland,  Oregon,  at  the 
rate  of  14  miles  per  hour  and  returns  at  the  rate  of  21 
miles  per  hour,  making  the  trip  in  6J  hours.  What  is  the 
distance  between  the  two  cities  ? 

15.  A  machine  running  between  Everett  and  Tacoma, 
Washington,  averaged  14  miles  per  hour  going  and  24^ 
miles  per  hour  returning,  making  the  trip  in  8^  hours.  Find 
the  distance  between  the  two  places. 

16.  A  car  averages  13^  miles  per  hour  for  7  hours.  If 
one  gallon  of  gasolene  will  carry  the  car  11|  miles,  how  many 
gallons  will  be  used  on  the  trip  ? 


36  APPLICATIONS  OF  ALGEBRA 

17.  If  it  requires  one  gallon  of  gasolene  to  run  a  car  11-|- 
miles,  how  many  gallons  will  be  required  to  run  the  car 
from  San  Francisco  to  Los  Angeles  (472.2  miles)  ?  Find 
the  cost  of  the  gasolene  at  15  cents  per  gallon. 

18.  How  many  revolutions  of  the  fly-wheel  are  made  per 
minute  if  the  car  goes  13^  miles  per  hour,  the  gear  ratio  is 
Tj\,  and  the  circumference  of  the  rear  wheel  is  9  ft.  ? 

19.  How  many  revolutions  per  minute  will  be  made  by 
the  fly-wheel  of  a  motor-car  travelling  from  Long  Beach  to 
Los  Angeles  (21  miles)  in  42  minutes,  if  the  gear  ratio  is  ^^ 
and  the  circumference  of  the  rear  wheel  is  8  ft.  ? 

20.  In  a  Cadillac  car  gear  F  has  17  teeth,  A  31,  B  24, 
G  31,  C  17,  H  31,  D  13,  E  13,  I  15,  and  J  55. 

(a)  The  final  gear  ratio  for  direct  drive,  or  high  speed, 

is   -.      Find  X.      (See  Section  15.) 
X 

(b)  Find  the  gear  ratio  for  intermediate  speed. 

(c)  Find  the  gear  ratio  for  low  speed. 

(d)  Find  the  gear  ratio  for  reverse  speed. 

21.  The  fly-wheel  of  an  auto-car  makes  900  revolutions 
per  minute,  the  gear  ratio  is  3^,  the  diameter  of  the  rear  wheel 
is  3  ft.     Find  the  distance  the  car  travels  in  10  minutes. 

22.  A  machine  with  intermediate-speed  gear  ratio  -^ 
travels  1093  ft.  per  minute.  How  many  revolutions  per 
minute  is  the  fly-wheel  making  if  the  diameter  of  the  rear 
wheel  is  36  in.  ? 

23.  A  motor-car  runs  forward  with  high  gear  ratio  -^  for 
1  minute;  the  fly-wheel  turns  800  times  per  minute,  and  the 
circumference  of  the  rear  wheel  is  9  ft.  How  many  minutes 
will  it  take  to  back  up  to  the  starting-place,  with  the  reverse 
gear  ratio  -5^  and  the  fly-wheel  revolving  at  the  same  rate 
as  before  ? 


AUTOMOBILES  IN  MOTION  37 

It  has  been  found  by  trial  that  on  a  level  macadamized 
road,  in  order  to  keep  a  car  in  motion,  it  requires  an  average 
force  or  pull  of  25  lbs.  for  each  1000  lbs.  of  the  weight  of  the 
car. 

If  F  =  average  force  required  to  keep  the  car  in  motion, 
and    W  =  total  weight  of  the  car  in  pounds, 

F         25  1  W 

^'^^^  IP  =  1000  =  40'°' ^  =  40- 

(In  the  following  problems  the  resistance  of  the  air  is  neg- 
lected.) 

The  number  of  foot-pounds  of  energy  necessary  to  keep  the 
car  moving  depends  not  only  upon  the  weight  of  the  car  but 
also  upon  its  speed,  which  should  be  expressed  in  feet  per 
minute. 

If  S  is  the  speed  in  miles  per  hour, 

Cr 

then         rr:  is  the  speed  in  miles  per  minute, 

5280  <S 
and  — WF. —  or  88  S  is  the  speed  in  feet  per  minute.    Why  ? 

The  number  of  foot-pounds  of  energy  per  minute  used  by 
a  traveUing  car  is  the  product  of  force  and  speed,  or 

W  11  WSi 

E  =  FS  =  ^XSSS==  i^^ 


Since  one  horse-power  is  33,000  ft. -lbs.  per  minute,  the 

g  ca 

TVS 


horse-power  used  m  a  travelhng  car  is  — = —  X  »^  ^^^,  or 

o  Ot5,uOU 


HP  = 


15,000 


24.  The  weight  of  a  loaded  automobile  is  3250  lbs.  What 
horse-power  (HP)  is  necessary  to  drive  it  over  a  smooth, 
level  road  at  the  rate  of  30  miles  per  hour? 

25.  An  auto-car  weighing  4200  lbs.  uses  9.8  horse-power 
in  speeding  over  a  level  macadamized  road.  How  fast 
does  it  go  ? 


38 


APPLICATIONS  OF  ALGEBRA 


26.  A  motor-car  uses  10.4  horse-power  while  going  at 
the  rate  of  40  miles  per  hour  over  a  smooth,  level  road. 
What  is  the  weight  of  the  car  ? 

27.  An  automobile  camping-trip  along  the  coast  of  Cal- 
ifornia shows  the  following  record.  The  meter  read  8852 
miles  when  the  party  started  from  Los  Angeles. 


STARTED 

ARRIVED 

METER 
READ- 
ING 

PLACE* 

TIME 

PLACE 

TIME 

1 

2 
3 
4 
5 
6 
7 

8 

9 

10 

11 

Los  Angeles.. . 

Camp 

Camp 

Camp 

Camp 

Camp 

Side  trip 
(Monterey) . . 

Camp 

Camp 

Camp 

Camp 

July 

19,  3.05  P.M. 

20,  6.45  A.M. 

21,  7.25  A.M. 

22,  9.30  A.M. 

23,  8.03  A.M. 

24,  6.05  A.M. 

25,  9.20  A.M. 

26,  8.05  A.M. 

27,  9.25  A.M. 

28,  1.15  P.M. 

29,  8.50  A.M. 

San  Fernando 

Santa  Barbara 

T/Os  Olivos 

San  Luis  Obispo . . 
Jolon 

5.25  P.M. 
6.35  P.M. 
6.10  P.M. 
6.10  P.M. 
7.35  P.M. 
7.05  P.M. 

7.25  P.M. 
7.40  P.M. 

11.45  A.M. 

6.50  P.M. 
1.55  P.M. 
5.25  P.M. 

8876 
8970 
9045 
9105 
9160 
9246 

9306 
9398 
9417 

9445 
9470 
9515 

Salinas 

Salinas 

Ben  Lomond 

Redwood  Park... 
Divide  near 

Los  Gatos 

San  Jose 

Oakland 

The  expense  record  of  the  trip  was  as  follows:  July  19, 
axle,  etc.,  $9.35;  5  gallons  gasolene,  $1.25;  2  quarts  oil,  50 
cents.  July  22,  5  gallons  gasolene,  $1.25;  1  gallon  oil,  80 
cents.  July  23,  4  gallons  gasolene,  $1.00.  July  24,  8  gal- 
Ions  gasolene,  $2.00,  1  gallon  oil,  70  cents.  July  26,  8  gal- 
lons gasolene,  $2.00,  1  gallon  oil,  75  cents.  July  31,  1  casing 
(tire),  $25.30.  At  the  end  of  the  trip  there  were  6  gallons 
of  gasolene  left. 

(a)  How  long  was  the  trip  ? 

(b)  Find  the  average  speed  per  hour  for  each  day's  travel 
and  the  average  speed  per  hour  for  the  trip. 

(c)  Find  the  total  cost  of  the  maintenance  of  the  ma- 
chine, exclusive  of  fuel,  and  the  average  cost  per  day. 

(d)  Find  the  average  cost  of  gasolene  and  oil  per  day  and 
per  hundred  miles. 


AUTOMOBILES  IN  MOTION  39 

19.    Formulas  concerning  Automobiles. 

BEcnox 

Area  of  Piston:    a  =  ^  =   .7854^2 4 

4 


Force  and  Pressure:  F  =  Pa 


F  = 


Foot-pounds  of  Energy :  E  =      » 


4      J 

Pla 
12 


48 
Energy  per  Minute  per  Cylinder:    E  =  -^j-     ...       6 

Energy  per  Minute:   E  =  6 

Horse-Power  of  Engine:  HP  =  ^^  =  ^4  x  Ss'oOO  -      ^ 

„„  Plird^nc 

or  HP  = 


96X33,000 
also  HP  =  .224^  +  1)dc    an  i9i5) 

HP   =  |C?2C  an  1916) 

t        r' 

Gears  in  Mesh:    -.  —  -  ot  ir  —  t'r' 9 

r       r 

Revolutions  of  Pinion:  no.  rev.  =  -7-7 14 

h  h 

Final  Gear  Ratio  = — ^ i — i- 15 

rev.  ny-wneel 

Distance  Travelled:    d  =  nrcm 15 

Foot-Pounds  of  Energy  to  Keep     „  _  IIWS 

Car  Running:  5  *      * 

ws 

Horse-Power  of  Moving  Car:  HP  =  ^^  ^.^       ...     18 

lo,UUU 


^.^^jgg 

jggg^ 

14  DAY  USE                     i 

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